Abstract: In this paper it is proven the following conjecture: If $G$ is a subgroup of the permutation group $S_{n}$ and $M$ is a 2-dimensional real manifold, then $M^{n}/G$ is a manifold if and only if $G=S_{m_{1}}\times S_{m_{2}}\times \cdots \times S_{m_{r}}$ where $S_{m_{1}},\ldots ,S_{m_{r}}$ are permutation groups of partition of $\{1,2,\ldots ,n\}$ into $r$ subsets with cardinalities $m_{1},\ldots ,m_{r}$, and $M^n$ is the topological product of $n$ copies of $M$.
Keywords: permutation products on manifolds, cyclic products on manifolds, permutation group
Classification (MSC2000): 53C99, 20B35
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